Quantitative recurrence results

Summary LetX be a probability measure spaceX=(X, , ) endowed with a compatible metricd so that (X,d) has a countable base. It is well-known that ifTXX is measure-preserving, then -almost all pointsxX are recurrent, i.e., . We show that, under the additional assumption that the Hausdorff -measureH (X) ofX is -finite for some >0, this result can be strengthened: , for -almost all pointsxX. A number of applications are considered.Oblatum 24-II-1992 & 8-II-1993Supported in part by NSF-DMS-9003450     

Asymptotics and Bounds for Multivariate Gaussian Tails

Let {X_n, n 1} be a sequence of centered Gaussian random vectors in , d 2. In this paper we obtain asymptotic expansions (n ) of the tail probability P{X_n >t_n} with t_n a threshold with at least one component tending to infinity. Upper and lower bounds for this tail probability and asymptotics of discrete boundary crossings of Brownian Bridge are further discussed.     

Positive Periodic Solutions of N-Species Neutral Delay Systems

In this paper, we employ some new techniques to study the existence of positive periodic solution of n-species neutral delay system N _i = N _i (t)[ _i (t) — _ij (t)N _j (t) — b _ij (t)N _j (t— _ij (t))— c _ij (t)N _j (t— _ij (t))].As a corollary, we answer an open problem proposed by Y. Kuang.     

First Exit Time from a Bounded Interval for a Certain Class of Additive Functionals of Brownian Motion

Let (B _t) _t0 be standard Brownian motion starting at y, X _t = x + ^t ₀ V(B _s) ds for x (a, b), with V(y) = y if y0, V(y)=–K(–y) if y0, where >0 and K is a given positive constant. Set _ab=inf{t>0: X _t(a, b)} and ₀=inf{t>0: B _t=0}. In this paper we give several informations about the random variable _ab. We namely evaluate the moments of the random variables , and also show how to calculate the expectations . Then, we explicitly determine the probability laws of the random variables as well as the probability by means of special functions.     

Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices

Let be a real separable Banach space and {X, X _{n, m}; (n, m) N ²} B-valued i.i.d. random variables. Set . In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N ² is studied. There is a gap between the moment conditions for CLIL(N ¹) and those for CLIL(N ²). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence to be almost surely conditionally compact in B, where, for 0, 1 r 2, N ^r (, ) = {(n, m) N ²; n m n exp{(log n) ^r–1 (n)}} and (·) is any positive, continuous, nondecreasing function such that (t)/(log log t) is eventually decreasing as t , for some > 0.     

Central limit theorems for nonlinear functionals of stationary Gaussian processes

Let X=(X _t ,t) be a stationary Gaussian process on (, ,P), letH(X) be the Hilbert space of variables inL ² (,P) which are measurable with respect toX, and let (U _s ,s) be the associated family of time-shift operators. We sayYH(X) (withE(Y)=0) satisfies the functional central limit theorem or FCLT [respectively, the central limit theorem of CLT if in [respectively,], where

On the spectrum determined growth assumption and the perturbation ofC 0 semigroups

The spectrum determined growth property ofC ₀ semigroups in a Banach space is studied. It is shown that ifA generates aC ₀ semigroup in a Banach spaceX, which satisfies the following conditions: 1) for any >s(A), sup{R(;A) | Re}<; 2) there is a ₀>(A) such that , xX, and , fX ^*, then (A=s(A). Moreover, it is also shown that ifA=A ₀+B is the infinitesimal generator of aC ₀ semigroup in Hilbert space, whereA ₀ is a discrete operator andB is bounded, then (A)=s(A). Finally the results obtained are applied to wave equation and thermoelastic system.     

A dissipative transformation with a trivial tail-algebra

Summary In [1], an example was given of a measure-preserving dissipative transformation T in a -finite measure space (X, , ), such that T is conservative in the measure space (X, , ) where . Here we shall show that for this transformation we actually have R ={ØX}[].     

TheO(n3) algorithm for a special case of the maximum cost-to-time ratio cycle problem and its coherence with an eigenproblem of a matrix

Let twon×n matrices be given, namely a real matrixA=(a_ij) and a (0, 1)-matrixT=(t_ij). For a cyclic permutation=(i ₁,i ₂,...,i _k) of a subset of N={1, 2, ..., n} we define _A;T(), the cost-to-time ratio weight of, as . This paper presents an O(n³) algorithm for finding (A;T)=max _A;T(), the maximum cost-to-time ratio weight of the matricesA andT. Moreover a generalised eigenproblem is proposed.     

Time Difference on the Brownian Curve

Let X _t and Y _t be respectively the locations of the maximum and minimum, over [0, t], of a real-valued Wiener process. We establish limsup and liminf iterated logarithm laws for , the time difference between the maximum and the minimum, as well as for max(X _t, Y _t) and min(X _t, Y _t).     

Moyennes uniformes et moyennes suivant une marche aléatoire

Summary Let be a bounded function on such that converges towards l as n goes to infinity, uniformly with respect to m. Let {X _n} be a random walk on , not concentrated on a proper subgroup of Then, with probability 1, converges towards l as n goes to infinity. The result also holds for any countable abelian group instead of . Other modes of convergence are considered (Cesaro convergence of order >1/2). The Cesaro convergence of expressions such that (X _n) (X _n+1) is also investigated.     

Recurrence-Transience for Self-similar Additive Processes Associated with Stable Distributions

For any stable distribution on the line, recurrence-transience of the selfsimilar additive process {X _t ,t0} with (X ₁)= is determined. Comparison with the stable Lévy process {Y _t ,t0} with (Y ₁)= is made: if is not strictly stable, then {Y _t } is transient but {X _t } is recurrent except the obviously transient case of monotone sample functions.     

A remark on the strong law of large numbers

The strong law of large numbers for independent and identically distributed random variablesX _i ,i=1, 2, 3,... with finite expectationE|X ₁| can be stated as, for any >0, the number of integersn such that \varepsilon $$ " align="middle" border="0"> ,N is finite a. s. It is known thatEN < iffEX ₁ ² < and that ² EN var X₁ as 0, ifE X ₁ ² <. Here we consider the asymptotic behaviour ofEN (n) asn, whereN (n) is the number of integerskn such that \varepsilon $$ " align="middle" border="0"> andE N ₁ ² =.     

Separation of two (possibly unbounded) components of the spectrum of a linear operator

LetX be a complex Banach space andA: D(A)X a densely defined closed linear operator whose resolvent set contains the real line and for which (–A)^–1 is bounded onR. We give a necessary and sufficient condition, in terms of the complex powers ofA and –A, for the existence of a decompositionX=X ₊X _–, whereX _± are closed subspaces, invariant forA, the spectra of the reduced operatorsA _± are {(A);Im>0} and {(A);Im<0} respectively, and (–A _±)^–1 is bounded forIm0.Finally we give an example of an operator in anL ^p-type space for which the decomposition exists if 1<p<+ and does not exist ifp=1.     

Estimates for the Distributions of the Sums of Subexponential Random Variables

Let be a random walk with independent identically distributed increments . We study the ratios of the probabilities P(S _n >x) / P(₁ > x) for all n and x. For some subclasses of subexponential distributions we find upper estimates uniform in x for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence P(S > x) E P(₁ > x) as x . Here is a positive integer-valued random variable independent of . The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process.     

Investments in stochastic maximum flow networks

Each are (i, j) of the network has capacity _ij where _ij is a non-negative random variable. The capacity of any arc may be reduced increased by an amountu _ij 0 at a cost ofc _ij u _ij . The objective is to maximizev–Kc _ij u _ij wherev is the expected maximum flow. This problem is formulated as a two-stage linear program under uncertainty. Each feasible generates a constraint where is the probability arc (i, j) is in the minimum cut set and the expected value of the maximum flow under . The formulation is later generalized to include certain conditions under which the increase in capacity of an arc may be a non-deterministic function of the investmentc _ij u _ij .     

Couplings of markov chains by randomized stopping times

Summary We consider a Markov chain on (E, ) generated by a Markov kernel P. We study the question, when we can find for two initial distributions and two randomized stopping times T of (X _n ) _nN and S of ( X _n ) _nN , such that the distribution of X _T equals the one of X _S and T, S are both finite.The answer is given in terms of -, h with h bounded harmonic, or in terms of .For stopping times T, S for two chains ( X _n ) _nN ,( X _n ) _nN we consider measures , on (E, ) defined as follows: (A)=expected number of visits of ( X _n ) toA before T, (A)=expected number of visits of ( X _n ) toA before S.We show that we can construct T, S such that and are mutually singular and ( _v X _T )=( X _S . We relate and to the positive and negative part of certain solutions of the Poisson equation (I-P)(·)=-.     

Invariance principles for random walks on hypergroups on ℝ2 and ℕ

Let (X _n:n) be i.i.d. with finite variance and values in a hypergroupK:=₊ or and _j=1 ⁿ X _j be the randomized sum of these random variables. It is shown that the processes converge in distribution to a Gaussian process in the caseK=₊, that the processes converge towards a Bessel process on ₊ in the case of polynomial growth of the hypergroupK=₊ or , and that in the case of exponential growth converges towards a Brownian motion asn.     

Sur la convexité holomorphe des revêtements linéaires réductifs d’une variété projective algébrique complexe

Let X be a smooth connected compact projective variety over . We study the Shafarevich conjecture concerning holomorphic convexity along the lines of Kollárs approach, when the fundamental group of X admits large finite dimensionnal representations. We prove that, given n, the topological covering space of a projective algebraic compact complex manifold X corresponding to the intersection of the kernels of all linear reductive representations of ₁(X) to GL_n() is holomorphically convex. In the surface case, this is a corollary of a theorem due to Katzarkov and Ramachandran.      

Bounding the Maximal Height of a Diffusion by the Time Elapsed

Let X=(X _t ) _t0 be a one-dimensional time-homogeneous diffusion process associated with the infinitesimal generator